Except for a few types of bills that customarily originate in the house, most bills are introduced roughly simultaneously in both houses so that the information for debating the bill doesn't have to be brought twice. This obviously doesn't guarantee a bill will pass because it is required to pass both houses.
Every bill has to go through both chambers, but they usually originate in one and then are passed to the other once the originating chamber affirms them.
It is not common to push two independent bills simultaneously, despite your assertion so.